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Sommerfeld's Velocity Addition in 2D Special Relativity

I presented yesterday at the APS april meeting and it was lots of fun!  The room was fairly packed and there were a number of other interesting talks in the session that was on the history of physics.  One of the most interesting presentations discussed how Tesla viewed theories of the luminiferous ether.

During my talk, I discussed how the special relativistic addition of velocities, in one dimension at least, is very simple if you first express the velocities in terms of rapidities.

As it turns out, the addition of velocities in two dimensions is much more difficult until you understand what's going on.  It seems mysterious at first, but it's not.  Simply put, the problem is that if you add a velocity in the east direction with a velocity in the north direction, you'll get a different answer than you will if you add a velocity in the north direction followed by one in the east direction.

That's right, change the order that you add velocities in and you get a different answer.  Duhm dum Duuhhhmm!!!  (Hopefully that came off as a suspenseful noise)  This appears so mysterious at first that we gave it a somewhat perplexing, yet impressive sounding definition:

When two non-collinear boosts are added, they add to the velocity of the moving object and provide an additional rotation of its direction in space.
To add to the mystery, we as physicists went off and named the angle that the object rotates through.  We even gave it more than one name, sometimes we call it the Thomas angle, (associated with the Thomas precession), and sometimes we call it the Wigner angle, (after a famous physicist who put lots of things in physics in terms of groups).  You'll also hear the term parallel transport of vector fields thrown around.

All this impressive sounding jargon refers to something that everyone who knew spherical trigonometry at the start of the 20th century knew about, (this was a lot of people since spherical trig is the tool used for ship navigation on the earth).  If we decide to steer along the equator of the earth for say 1000 kilometers due east, and then turn north for 500 kilometers, we'll wind up in a different spot than if we'd first steamed along for 500 kilometers due north first and then 1000 kilometers due east.

The whole thing amounts to little more than calculating the circumference of circles with different radii.  No matter where I am on the earth, I can draw a circle pointed due east that will eventually make it all the way around, bumping me in the butt on its return.  The full circle will have 360 degrees of longitude.  If I happen to be at the equator of the earth, I have to travel pi times the diameter of the earth at the equator to get all the way back around.  However, if I'm really close to the north pole, and I head due east, I only have to travel a few feet to make a full circle.

OK, here's an example of how adding two different distances on a sphere in different orders gets you different answers.  If I'm at the equator of the earth, I'm on the largest circle I can find.  If I head due east and then north, I wind up at the orange start on the map:

But..... if I head north then east, I'll move the same amount north, but much further around the circle at that latitude, (because the circle has a much smaller radius), so I'll wind up about where the light blue star is:

And so it goes.  As it turns out, velocities in different directions in special relativity add like the math is being done on the surface of a sphere.  The math is a little abstract, certainly.  The radius of the sphere is negative, the sphere is four dimensional, and the value of the radius is the speed of light.  The math works out the same though!  Sommerfeld[1], the dapper looking guy in the last picture was one of the first to put this together, (if not the first).

1.  Sommerfeld's magnificent EM book:


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